It depends a great deal on the snow conditions. If the snow is deep and soft, you’re going to go all the way through no matter what. The more packed-down it is, the less you’ll break through and the easier the running will be. In general, snowshoe running is done on packed trails, rather than virgin snow. When I’m setting my own trail, I tend to run back and forth a bit to flatten it down, rather than going for a single pass.
“Normal” snowshoes will be larger and heavier and will be more likely to keep you on top of virgin snow, assuming it’s dense enough. You wouldn’t want to run in them because it would be much more tiring and you’d be far more likely to bang your ankles. They’re a lot less fun. 
Yes, your shoes will get wet, mostly from the snow melting on them. It’s good to wear wool socks, which retain warmth even when they get wet. Since you’re running, you’re not likely to get cold while you’re moving, but it can happen pretty quickly afterward when you stop.
Yes, definitely, though if you weren’t planning to run at all, larger ones would keep you more on top of the snow.
Yes, sorry. So the final price would be more like $200 to $230. I’m not entirely certain if there’s tax and shipping yet, but would work that in if necessary.
You’re on my list!